I'm quite sure that in a solo challenge the asked maximum exists / is finite. You need opponents to increase the value of Pirate Ship. Still, you should be able to reach at least 50.000 even with the treasure limit. Not sure whether you want to focus on actions: discard with Vault, Storeroom, Secret Chamber and then draw deck with Scrying Pool + Madman for ~$500 each time, or on treasures: play Black Market, all treasures, gain Mandarin with Horn of Plenty, draw deck with Madman. This should give you at least $2000 each time (include bank, some other kingdom treasures, and play coppersmith a lot, maybe diadem is useful), but the limit on treasures really hurts here. Probably you want to do both, but maybe the kingdom slots for vault/storeroom/sc are better used as kingdom treasures (to boost bank).

As with pretty much every Dominion puzzle, I'm pretty sure that the solution is a transformation of this.

It depends what you mean with "transformation". But I think the solution is totally different: you don't need to be fast (4 coins per turn is nothing), you want to have a completely different kingdom (probably only Black Market, Procession and King's Court in common with it), you want to use completely different tricks (as I mentioned above). I do not want to undersell CC's solution, it's awesome, but saying that every puzzle is similar to that one (at least, I assume that's what you mean) is not true.

Infinite. I play 5 pirate ships each turn, and my opponent gets 5 treasures back each turn through rouge. The minimum for the amount of treasure loss per turn would have to be much higher, at least around 30.

On your first turn you can't get more than $6. On later turns you can produce more money, so the intention was to force people to find a balance between the amount of money they produce and the length of time it takes to set up.

Infinite. I play 5 pirate ships each turn, and my opponent gets 5 treasures back each turn through rouge. The minimum for the amount of treasure loss per turn would have to be much higher, at least around 30.

The idea with the solo challenges was that you actually carry them out in one of the simulators, in a single player game. In any case, there aren't supposed to be any opponents.